Formula for Alexander polynomials of simple-ribbon knots derived.
problem Determining Alexander polynomials for simple-ribbon knots.
method Introduced simple-ribbon fusions and used them to derive a formula for Alexander polynomials.
result Formula for Alexander polynomials of simple-ribbon knots derived.
Study knots that divide ribbon knotted surfaces, computing their half ribbon genus and fusion number.
problem Understanding knots that divide ribbon knotted surfaces and their properties.
method Defining half ribbon knots, computing half ribbon genus and fusion number, and comparing with Levine-Tristram signatures.
result Computed half ribbon genus and fusion number for various knots, including new computations of doubly slice genus.
The paper calculates ribbon numbers for 12-crossing knots using Alexander polynomials.
problem Determining the minimum number of ribbon singularities for knots.
method Using Alexander polynomials and systematic treatment of knot invariants.
result Computed ribbon numbers for many 12-crossing knots.
Knot modules are linked to ribbon 2-knots with geometric constraints.
problem Characterizing knot modules of ribbon 2-knots.
method Showing every Z-torsion free knot module can be realized by a ribbon 2-knot with specific geometric properties. result Knot modules of ribbon 2-knots have geometric constraints with group of geometric dimension at most 2.
Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.
problem Determining Alexander polynomials for ribbon and virtual knots.
method Using ribbon's intrinsic singularity information, defining half Alexander polynomial, and developing simplified formulas.
result New formulas for Alexander polynomials of general knots and virtual knots in terms of Gauss diagrams.
Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.
Study shows not all ribbon knots can be symmetric unions.
problem Whether every ribbon knot can be a symmetric union.
method Exhibited a specific ribbon Montesinos knot that cannot be a symmetric union.
result Found a ribbon knot that is not a symmetric union.
Two-bridge ribbon knots have symmetric union presentations.
problem Characterizing two-bridge ribbon knots.
method Symmetric union presentations and partial knot analysis.
result Symmetric union presentations for various two-bridge ribbon knots.
Axel Seeliger tabulated ribbon knots and found non-symmetric examples.
problem Identifying non-symmetric ribbon knots.
method Tabulated symmetric union presentations and band diagrams for ribbon knots.
result Found non-symmetric ribbon knots with 11 and 12 crossings.
Sharp knots and iterated cables lead to ribbon knots or failure of slice-ribbon conjecture.
problem Understanding the concordance properties of knots and their cables.
method Using Seifert genus and concordance invariant γ0 from bordered Heegaard Floer homology.
result Connected sums of γ0-sharp fibered knots are ribbon only if they are of a specific form, or the slice-ribbon conjecture fails.
We consider ribbon n-knots for n\geq 2. For such knots we define a set of moves on ribbon disks, and show that any two ribbon disks for isotopic knots are related by a finite sequence of such moves and ambient isotopies. Using this we are able to prove that there is a natural geometric correspondence between ribbon n-k…
This paper investigates symmetric ribbon numbers of low-complexity knots.
problem Determining the minimum number of ribbon singularities in symmetric ribbon disks for knots with up to 12 crossings.
method Systematic investigation using knot polynomials and determinants.
result Novel lower bounds for symmetric ribbon numbers of knots with up to 12 crossings.
New knots bound multiple non-isotopic ribbon disks.
problem Finding knots that bound multiple non-isotopic ribbon disks.
method Classification of fibered, homotopy-ribbon disks for generalized square knots.
result Infinitely many knots bound infinitely many pairwise non-isotopic ribbon disks.
Study ribbon concordance and minimal compressions, proving new results about fibered knots.
problem Understanding ribbon concordance and minimal compressions of surface homeomorphisms.
method Proving monotonicity of simplicial volume and dilatation under ribbon concordance, algorithmic enumeration of minimal compressions.
result Every fibered knot has only finitely many predecessors in the ribbon-concordance partial order.
New knot quandles distinguish ribbon knots with isomorphic groups.
problem Distinguishing knots with isomorphic fundamental groups.
method Examined knot quandles of Suciu's ribbon knots and computed their types.
result Knot quandles of Suciu's ribbon knots are mutually non-isomorphic.
New bounds found for ribbon numbers of knots and links.
problem Finding bounds for the ribbon number of knots and links.
method Using determinant and Jones polynomial, we find new lower bounds and prove the finiteness of certain sets.
result We determine the set of Jones polynomials for ribbon knots with 11 or fewer crossings.
Researchers found a linear upper bound on ribbonlength for torus and twist knots.
problem Finding the minimal ratio of length to width of flat knotted ribbons.
method Using grid diagrams and ribbon folding techniques to construct and analyze flat knotted ribbons.
result Proved a linear upper bound on ribbonlength for all non-trivial torus and twist knots.
The study limits the number of ribbon concordant fibered knots.
problem Understanding the relationship between ribbon concordance and fibered knots.
method Combining Floer homology results with fixed points of monodromy and volume inequalities.
result There are only finitely many hyperbolic fibered knots ribbon concordant to any given knot.
Ribbon knots with isomorphic quandles are stably equivalent.
problem Determining when ribbon knots with isomorphic quandles are equivalent.
method Using quandle isomorphism to show stable equivalence after finite connected sums.
result Ribbon knottings with isomorphic quandles are stably equivalent.
Injective map proven in knot Floer homology for strong homotopy-ribbon concordances.
problem Injectivity of knot Floer homology maps under strong homotopy-ribbon concordances.
method Knot Floer homology and properties of strongly homotopy-ribbon concordances.
result Injective map proven in knot Floer homology for strong homotopy-ribbon concordances.
Knots can be ordered by ribbon concordance, solving a long-standing question.
problem Ordering knots by ribbon concordance.
method Representation varieties of knot groups to SO(N) and relations induced by ribbon concordance. result Ribbon concordance forms a partial ordering on the set of knots.
New tangles found for 21 ribbon knots with 10 or fewer crossings.
problem Characterizing ribbon knots through tangle forms.
method For each of 21 ribbon knots, found a 4-strand tangle that satisfies specific closure properties.
result Each of 21 ribbon knots can be represented by a tangle that meets the specified closure conditions.
Knot Floer homology detects ribbon concordance.
problem Detecting ribbon concordance between knots.
method Using knot Floer homology to study ribbon concordance.
result The map induced by ribbon concordance on knot Floer homology is injective.
New insights into knot fusion numbers via cabling.
problem Understanding fusion numbers of ribbon knots and their behavior under cabling.
method Utilizing knot Floer homology and cabling formulas to analyze fusion numbers.
result The fusion number and strong homotopy fusion number of (p,1)-cable knots are preserved.
We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2-knots. As an application, we produce infinite families…
Prove strong ribbon concordance induces a partial order on links, certify minimality for a handful of knots, and find minimal ribbon minimal knots.
problem Prove strong ribbon concordance induces a partial order on links.
method Use results from knot Floer homology to certify minimality under the ribbon partial order.
result Certify minimality for a handful of knots and find minimal ribbon minimal knots.
Algorithm finds ribbon disks for alternating knots, resolving sliceness for most prime knots.
problem Finding ribbon disks for alternating knots to determine sliceness.
method Algorithm based on Donaldson's diagonalization theorem, computer implementation.
result Successfully finds ribbon disks for many prime knots, resolves sliceness for most.
Study knot distance using ribbon concordances and Khovanov homology.
problem Knot distance measurement.
method Lower bound construction using X-action on Lee's perturbation of Khovanov homology.
result Lower bound on ribbon distance between knots.
Ribbonness proven for slice knots, solving an old question.
problem Proving ribbonness for slice knots.
method Showing a link bounds a ribbon surface that is a renewal embedding of a bounding surface.
result Every slice knot is a ribbon knot.
Classifies fibered ribbon pretzels, except for a few cases.
problem Classifying fibered ribbon pretzel knots up to mutation.
method Combining lattice embedding techniques with Gabai's classification of fibered pretzel knots, and exhibiting ribbon disks.
result Complete classification except for a few cases.
New theorem connects handle-ribbon knots to slice derivatives.
problem Characterizing knots that bound handle-ribbon disks.
method Proves equivalence between handle-ribbon and slice derivatives.
result Handle-ribbon knots are equivalent to slice derivatives.
Study shows knots with similar Blanchfield forms can be homotopy ribbon concordant.
problem Understanding homotopy ribbon concordance for knots.
method Using Blanchfield pairings and twisted Alexander polynomials.
result Existence of infinite families of knots with same Blanchfield form but not homotopy ribbon concordant.
Khovanov homology shows (4,5) torus knot is a summand in its concordance class.
problem Understanding the global ribbon minima in knot concordance classes.
method Application of Khovanov homology to ribbon concordance.
result Khovanov homology of (4,5) torus knot is a summand in any knot's homology.
This paper gives a new obstruction for ribbon-move equivalence of 2-knots. Let K and K′ be 2-knots. Let K and K′ are ribbon-move equivalent. One corollary to our main theorem is as follows. A 2-dimensional fibered knot whose fiber is the punctured 3-dimensional torus is not ribbon-move equivalent to any 2-dimen…
This paper confirms a bound for folded ribbonlength of 2-bridge knots.
problem Bounding the folded ribbonlength of 2-bridge knots.
method Investigated the folded ribbonlength of 2-bridge knots and proved a linear upper bound.
result The folded ribbonlength of a 2-bridge knot K is bounded above by 2c(K)+2. New condition for Alexander polynomial factorization in 4D knots.
problem Alexander polynomial factorization for 2-knots in S4. method Alternative notion of ribbon 2-knots to prove factorization condition.
result Topological condition for factorization of Alexander polynomial.
Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.
problem Understanding the symmetric braid index of ribbon knots.
method Defining symmetric braid index, using Khovanov homology, and calculating bounds.
result Existence of knots with symmetric braid index greater than braid index.
Wide ribbons of large width can connect knots of different types.
problem Understanding how knots are connected by wide ribbons.
method Analyzing the outer ribbon edge of wide ribbons associated with knots.
result For large width, the outer ribbon edge of a knot typically converges to a finite set of knot types.
Machine learning finds knots that bound ribbon disks.
problem Detecting ribbon knots in topology.
method Bayesian optimization and reinforcement learning.
result Successfully detected many ribbon knots up to 70 crossings.
The twisting number of a ribbon knot is at least as large as its doubly slice genus.
problem Proving a lower bound for the twisting number of ribbon knots in terms of their doubly slice genus.
method Analyzing symmetric unions and tangle replacements to establish the bound.
result Ribbon knots have arbitrarily high twisting numbers, matching their doubly slice genus.
A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.
A pretzel knot K is called odd if all its twist parameters are odd, and mutant ribbon if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice, odd, 5-stranded pretzel knots are mutant ribbon. We d…
Revised proof shows ribbonness of surface-links in 4-sphere.
problem Determining ribbonness of surface-links in 4-sphere.
method Surgery along fusion 1-handle systems to prove ribbonness.
result Surface-links in 4-sphere are ribbon if certain conditions are met.
New restrictions found on triple linking numbers of knot derivatives.
problem Understanding triple linking numbers of knot derivatives.
method Defined a new obstruction for Z[Z]-homology ribbon knots.
result New non-doubly slice knots discovered.
Positive knots are minimal in a specific knot ordering.
problem Understanding the minimal structure of positive knots in a specific knot ordering.
method Analyzing ribbon concordance and using properties of rational Alexander modules.
result Positive knots are minimal in the ribbon concordance ordering.
The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.
problem Proving the non-triviality of welded knots and ribbon torus-knots.
method By generating examples and determining the fundamental group of the concerned welded knot.
result Non-triviality of welded knots and ribbon torus-knots is demonstrated.
Survey on folding knots in paper to minimize length.
problem Minimizing the ribbonlength of folded ribbon knots.
method Review of Kauffman's model and upper bounds using crossing number.
result Upper bounds of ribbonlength for twist and torus knots.
Ribbon cobordism forms a partial order in 3-manifolds.
problem Understanding partial orders in 3-manifolds.
method Utilizing recent methods from Ian Agol's work on knot concordance.
result Ribbon rational homology cobordism forms a partial order.